Active error correction for Abelian and non-Abelian anyons

We consider a class of decoding algorithms that are applicable to error correction for both Abelian and non-Abelian anyons. This class includes multiple algorithms that have recently attracted attention, including the Bravyi-Haah RG decoder and variants thereof. They are applied to both the problem of correcting a single burst of errors (with perfect syndrome measurements) and active correction of continuously occurring errors (with noisy syndrome measurements). For Abelian models we provide a threshold proof in both cases for all decoders in this class, showing that they can arbitrarily suppress errors when the noise rate is under a finite threshold. For non-Abelian models such a proof is found for a single burst of errors. The reasons why the proof cannot be applied to the case of continuously occurring errors are discussed.

Figure 1

Examples of errors, and how these may be dealt with by a decoder. Here only a one-dimensional slice of the spatial plane is considered, represented horizontally. Time corresponds to the vertical direction, with time flowing in the upwards direction. Points at which anyons are measured are denoted by orange circles. These are filled if the anyon is unexpected, and so correspond to a defect. Points at which no anyon was measured when one was expected, another defect, are denoted by pink circles. Red lines denote errors (horizontal for spin and vertical for measurement), while green lines denote attempted anyon movement by the decoder. (a) A chain of three spin errors creates a pair of anyons. For a few time steps it is most likely that the anyons are simply due to measurement errors. After this they are most likely to have been created in a pair, and so are moved towards each other until annihilation. (b) Same as before, except that an additional spin error moves the right anyon. This creates an additional pair of defects. The first movement operation applied to the left anyon also fails, meaning that the anyon does not move as expected. This also creates an additional pair of defects. Nevertheless, these effects are accounted for, and the anyons are finally annihilated. (c) Two examples of measurement errors: one where no anyon is expected and one where one is. Both lead to a pair of defects. (d) A chain of spin errors create three anyons, though one is hidden for a time by measurement errors. The decoder first pairs those it can see, but they do not annihilate. The final anyon, once visible, is annihilated with the fusion product.

Figure 2

Errors and correction for the case described in text. Errors are shown in red, and the path of anyons is shown in green. Anyon locations at the time of the later error are shown in blue, with initial positions of anyons created by the earlier error in light blue.

Figure 3

Square lattice on which a quantum double model is defined is shown in gray. The corresponding hexagonal code lattice is shown in blue. Spins are located on the edges of the former, and vertices of the latter.